I Introduction
Alongside the twouser interference channel [1, 2, 3, 4], the twouser XChannel is a canonical example to study the impact of interference in communication networks. In the XChannel, each transmitter has a common message intended for both receivers as well as a private message intended for each receiver. This problem has been studied extensively in the literature and several interference management techniques have been proposed [5, 6, 7]. For instance under instantaneous channel state information (CSI) model, it was shown in [6] that interference alignment can provide a gain over baseline techniques (e.g.
, orthogonalization). This gain is expressed in terms of degreesoffreedom (DoF) which captures the asymptotic behavior of the network normalized by the capacity of the pointtopoint channel when power tends to infinity.
Attaining instantaneous channel state information at the transmitters (CSIT) in many realworld scenarios may not be feasible. In such cases, a more realistic model is the delayed CSIT in which by the time the CSI arrives at the transmitters, the channel has already changed to a new state. Under the delayed CSIT model, authors in [8] developed a scheme that achieves DoF. Later, it was shown that if we limit ourselves to linear encoding functions, then is indeed the optimal DoF [9]. These results provide valuable insight into the behavior of XChannels. However, in information and communication theory, the ultimate goal is to understand the behavior of wireless networks for any signaltonoise ratio (SNR). In other words, we are interested in capacity results rather than DoFtype results. Moreover, while one might argue that most practical communication protocols are linear, limiting the encoding functions to be linear removes the majority of potential encoding functions and from an informationtheoretic perspective, this is not desirable. Finally, authors in [8] and [9] study a subset of XChannels in which transmitters only have private messages for the receivers and the issue of common messages in XChannels is not addressed, and as we will show, including common messages introduces new challenges.
In this work, we address these issues by deriving the capacity region of XChannels with delayed CSIT and common messages under a finitefield fading model introduced in [10]
. In this model, the channel gains at each time are drawn from the binary field according to some Bernoulli distribution. The inputoutput relation of this channel model at time
is given by(1) 
where , the channel gains are in the binary field, is the transmit signal of transmitter at time , and is the observation of receiver at time . All algebraic operations are in . In the delayed CSIT model, we assume that the transmitters at time have access to
(2) 
The XChannel poses several new challenges when compared to the interference channel. In the interference channel, each transmitter has a private message for its corresponding receiver and to maximize the overall achievable rate, each transmitter tries to minimize the interference subspace at the unintended receiver. In the context of XChannels, however, each transmitter has a private message for each one of the receivers which changes the interference dynamics of the problem since receivers are now interested in signals of both transmitters. On top of this, each transmitter has to deliver a common message to the receivers. In this work, we establish the capacity region of finitefield fading XChannels under the delayed CSIT assumption. We derive a new set of outerbounds for this problem. We also propose a distributed transmission strategy that harvests the delayed CSI to combine and to repackage previously communicated signals in order to deliver them efficiently. We show that this transmission strategy matches the outerbounds, thus, characterizing the capacity region.
To derive the outerbounds, we rely on an extremal entropy inequality to capture both the impact of delayed CSIT and the challenge of delivering the common messages. This inequality quantifies the ability of a transmitter to favor one receiver over the other in terms of provided entropy when: both receivers need to obtain some common entropy, and the transmitter has access to the delayed channel state information. In particular, this extremal inequality quantifies such that the following inequality holds for any input distribution:
(3) 
where indicates the common message, and is the private message for . Using (3) and a genieaided argument, we obtain the outerbounds.
To achieve the outerbounds, we treat the XChannel as a combination of a number of wellknown problems for which the capacity region is known. By adjusting different rates for the XChannel, we can recover several other problems such as the interference channel, the multicast channel, the broadcast channel, and the multipleaccess channel. We demonstrate how to utilize the capacityachieving strategies of such problems in a systematic way in order to achieve the capacity region of the XChannel. We show that, however, if we treat the XChannel as a number of disjoint subproblems, we will not achieve the capacity and in some regimes, we need to interleave the capacityachieving strategies of different subproblems and execute them simultaneously.
Ii Problem Formulation
We consider the twouser Binary Fading XChannel as illustrated in Fig. 1 with two transmitters and two receivers^{1}^{1}1In this work, we focus on the binary field. Extending the results from the binary field to larger Galois fields is possible and rather straightforward.. In the binary fading model, the channel gain from transmitter to receiver at time is denoted by , . The channel gains are either or (i.e.
), and they are distributed as independent Bernoulli random variables (independent across time and space). We consider the homogeneous setting in which
(4) 
for , and we define .
At each time , the transmit signal of is denoted by , and the received signal at is given by
(5) 
where the summation is in , and .
We define the channel state information (CSI) at time to be the quadruple
(6) 
and for natural number , we set
(7) 
where is defined in (6). Finally, we set
(8) 
In this work, we consider the delayed CSIT model in which at time each transmitter has the knowledge of the channel state information up to the previous time instant (i.e. ) and the distribution from which the channel gains are drawn (i.e. ), . Since receivers only decode the messages at the end of the communication block, without loss of generality, we assume that the receivers have instantaneous knowledge of the CSI.
For the XChannel, we consider the scenario in which , , wishes to reliably communicate

message to both receivers,

message to ,

and message to ,
during uses of the channel. We assume that the messages and the channel gains are mutually independent and the messages are chosen uniformly at random.
For transmitter , let messages and be encoded as using the encoding function , which depends on the available CSI at , see Fig. 1. Receiver is interested in decoding and given by
(9) 
and it will decode the messages using the decoding function :
(10) 
An error occurs when
(11) 
The average probability of decoding error is given by
(12) 
where the expectation is taken with respect to the random choice of messages.
We define
(13) 
A rate tuple is said to be achievable, if there exists encoding and decoding functions at the transmitters and the receivers respectively, such that the decoding error probabilities go to zero as goes to infinity. The capacity region is the closure of all achievable rate tuples.
Iii Main Results
In this section, we present the capacity region of the twouser Binary Fading XChannel under the delayed CSIT assumption, and we provide some technical insights and interpretations of the main results.
Iiia Statement of the Main Results
The following theorem establishes the capacity region of the twouser Binary Fading XChannel with private and common messages under the delayed CSIT assumption.
Theorem 1.
The capacity region is described by two sets of outerbounds. The first set is referred to as the Broadcast Channel (BC) bounds. These bounds describe the capacity region of the broadcast channel formed by one of the transmitters and the two receivers assuming the other transmitter is eliminated. These bounds can be thought of as the generalization of the results in [11, 12, 13] for the twouser case to include common messages. The second set is referred to as the XChannel (XC) bounds. We note that adding the two BC bounds results in
(16) 
which is dominated by the XC bounds since for any , we have
(17) 
The derivation of the outerbounds relies on an extremal entropy inequality that quantifies the ability of each transmitter in favoring one receiver over the other in terms of the available entropy subject to two constraints: both receivers need to obtain a baseline entropy (to capture the common messages), and transmitters have access to the delayed CSI. This inequality characterizes the limit to which the unwanted subspace at one receiver can be scaled down while the desired subspace at the other receiver is maximized. We use this inequality and a genieaided argument to derive the new outerbounds.
The twouser XChannel can be thought of as a generalization and a combination of several wellknown problems. For instance, if and are the only nonzero rates, then the problem is equivalent to the multipleaccess channel formed at , and if and are the only nonzero rates, then the problem is equivalent to the broadcast channel formed by . We demonstrate how to utilize the capacityachieving strategies of other problems, such as the interference channel and the multicast channel, in a systematic way in order to achieve the capacity region of the XChannel. We show that, however, if we treat the XChannel as a number of disjoint subproblems, we will not achieve the capacity in some regimes. In fact, in such regimes, we need to interleave the capacityachieving strategies of different subproblems and execute them simultaneously.
IiiB Illustration of the Main Results
To illustrate the results of Theorem 1, we focus on the symmetric case for , i.e.
(18) 
Fig. 2 depicts the twodimensional region of for . In this figure for a given , we define
(19) 
An interesting observation is that the sum of and (i.e. ) determines the size of the region rather than the individual values. For instance, Fig. 2(c) is the same for , , and . Moreover, as increases, the symmetric capacity, , decreases. The reason is that providing more common entropy to the receivers reduces the ability of each transmitter in favoring one over the other in terms of available entropy. In other words, providing more common entropy to the receivers reduces transmitters’ ability in performing interference alignment.
IiiC Comparison to the Interference Channel
For the twouser Binary Fading Interference Channel [10], there is no common message (i.e. ), and each transmitter only has a message for one receiver (i.e. ). Fig. 3 depicts the capacity region of the XChannel for which includes the capacity region of the interference channel. We note that in the XChannel, individual rates are limited by the capacity of the multipleaccess channel (MAC) formed at each receiver (i.e. ), whereas in the interference channel the limit is the capacity of the pointtopoint channel (i.e. ). Moreover, for some values of , the symmetric capacity of the XChannel is strictly larger than that of the interference channel. This issue is further discussed in Section VC and Fig. 8.
IiiD The Broadcast Channel Bounds
So far, we focused on and and as a result, the BC bounds did not play a role. As mentioned earlier, the BC bounds describe the capacity region of the broadcast channel formed by one of the transmitters and the two receivers assuming the other transmitter is eliminated. Suppose we set and equal to (i.e. eliminating the second transmitter), and we focus on and . These rates correspond to and are governed by the BC bounds as depicted in Fig. 4.
Iv Converse Proof of Theorem 1
In this section, we provide the converse proof of Theorem 1. The proof of the BC bounds has some similarities to that of the XC bounds and when possible we omit the duplications in the proofs.
BC Bounds: We first derive the Broadcast Channel bounds, i.e.
(20) 
By symmetry, it suffices to prove (20) for . As mentioned before, this bound corresponds to the Broadcast Channel formed by when is eliminated. In our proof, this fact is captured by conditioning on and . We have
(21) 
where as ; follows from the independence of messages; follows from Fano’s inequality; holds since messages are independent of channel realizations; follows from Claim 1 below; follows the fact that is a function of ; holds since . Dividing both sides by and letting , we get
(22) 
Similarly, we can obtain
(23) 
Claim 1.
For the twouser Binary Fading XChannel with private and common messages under delayed CSIT assumption as described in Section II, we have
(24) 
The proof of Claim 1 follows step by step that of Claim 2 that we will present and prove below and thus omitted.
XC Bounds: As mentioned before, the XC bounds cannot be obtained from the BC bounds. However, the derivation resembles the one we provided for the BC bounds with some modifications. To derive the XC bounds, we have
(25) 
where as ; follows from the independence of messages; follows from Fano’s inequality; holds since messages are independent of channel realizations; follows from Claim 2 below; holds since
(26) 
Dividing both sides by and letting , we get
(27) 
Similarly, we can obtain
(28) 
Claim 2.
For the twouser Binary Fading XChannel with private and common messages under delayed CSIT assumption as described in Section II, we have
(29) 
Proof.
We first note that
(30) 
Thus, proving (29) is equivalent to proving
(31) 
We have
(32) 
where follows from the fact that all signals at time are independent of future channel realizations; holds since ; is true since transmit signal is independent of the channel realization at time ; holds since conditioning reduces entropy; holds since ; follows from the definition of ; is true since all signals at time are independent of future channel realizations;
follows from the chain rule and the nonnegativity of the entropy function for discrete random variables. ∎
This completes the converse proof of Theorem 1, and in the following section we present the achievability proof.
V Achievability Proof of Theorem 1
XChannels can be thought of as a generalization of several known problems such as interference channels, broadcast channels, multipleaccess channels, and multicast channels. In the previous section, we developed a set of new outerbounds for this problem. In this section, we show that a careful combination of the capacityachieving strategies for other known known problems will achieve the capacity region of the XChannel. However, in Section VC we show that if we treat the XChannel as a number of disjoint subproblems, we will not achieve the capacity and in some regimes, we need to interleave the capacityachieving strategies of different subproblems and execute them simultaneously.
To describe the transmission strategy, we first present two examples. The first example describes a symmetric scenario associated with Fig. 2(b) and the second example describes a scenario in which transmitters achieve unequal rates. After the examples, we present the general scheme.
Va Example 1: Symmetric SumRate of Fig. 2(b)
For this particular example, we treat the XChannel as three separate problems listed below at different times, and we show that this strategy achieves the capacity.

For the first third of the communication block, we treat the XChannel as a twouser multicast channel as depicted in Fig. 5(a) in which each transmitter has a message for both receivers. For the twouser multicast channel with fading parameter , the capacity region matches that of the multipleaccess channel formed at each receiver [10] and depicted in Fig. 5(a) as well.

During the final third of the communication block, we treat the XChannel as a twouser interference channel with swapped IDs in which wishes to communicated with , see Fig. 5(c). In the homogeneous setting of this work, the capacity region of this interference channel with swapped IDs matches that of the previous case and is depicted in Fig. 5(c).
Achievable Rates: We note that as the communication block length, , goes to infinity, so do the communication block lengths for each subproblem. Thus, during the first third of the communication block, we can achieve symmetric common rates arbitrary close to . Normalizing to the total communication block, we achieve which matches the requirements of (VA). From [10] we know that for the twouser binary fading interference channel with delayed CSIT and , we can achieve symmetric rates of . Normalizing to the total communication block, we achieve which matches the requirements of (VA). Finally, during the final third of the communication block we treat the problem as a twouser interference channel with swapped IDs in which we can achieve symmetric rates of . Normalizing to the total communication block, we achieve which again matches the requirements of (VA). Thus, with splitting up the XChannel into a combination of three known subproblems, we can achieve the capacity region described in Theorem 1.
VB Example 2: Unequal Rates
In the previous subsection we focused on a symmetric setting. Here, we discuss a scenario in which transmitters have different types of messages with different rates for each receiver. More precisely, we consider the region in Fig. 2(c) for , and
(34) 
In this case, we can think of the XChannel in this case as two subproblems that coexist at the same time as described below.

The Binary Fading Broadcast Channel from as in Fig. 6(a) in which a single message is intended for both receivers. For this problem, the capacity can be achieved using a pointtopoint erasure code of rate .

The Binary Fading Broadcast Channel from as in Fig. 6(b) with delayed CSIT in which the transmitter has a private message for each receiver. For this problem, the capacity region is given in [11, 12] and depicted Fig. 6(b). As described below, in order to be able to decode the messages in the presence of the broadcast channel from , we first encode and using pointtopoint erasure codes of rate , and treat the resulting codes as the input messages to the broadcast channel of Fig. 6(b).
Achievable Rates: At each receiver the received signal from is corrupted (erased) half of times by the signal from . As a result, when we implement the capacityachieving strategy of [11, 12], we only deliver half of the bits intended for each receiver. However, since we first encode and using pointtopoint erasure codes of rate , obtaining half of the bits is sufficient for reliable decoding of and . Thus, we achieve
(35) 
which again matches the requirements of (VB). At the end of the communication block, receivers decode and , and remove the contribution of from their received signals. After removing the contribution of , the problem is identical to the broadcast channel from as in Fig. 6(a) for which we can achieve a common rate of .
VC Transmission Strategy
The two examples we have provided so far demonstrate the key ideas behind the transmission strategy and dividing the problem into a number of known subproblems. Suppose we would like to achieve
(36) 
An important difference between the XChannel and the interference channel is the fact that in the latter scenario, the individual rates are limited by the capacity of a pointtopoint channel, i,e, . As a result, for the interference channel we have [10]:
(37) 
However, in the XChannel no such limitation exists, and we have
(38) 
The difference is depicted in Fig. 8 for . This means that if we naively try to use the capacityachieving strategies of the subproblems independently, we cannot achieve the capacity region of the XChannel. The key idea to over come this challenge is to take an approach similar to the one we presented in Section VB and run different strategies simultaneously as described below.
The strategy that achieves the rates in (VC) is similar to what we presented in Section VA. Define
(39) 
First suppose
(40) 
Then the transmission strategy is as follows.

For the first fraction of the communication block, we treat the XChannel as a twouser multicast channel as depicted in Fig. 7(a) in which each transmitter has a message for both receivers. For the twouser multicast channel with fading parameter , the capacity region matches that of the multipleaccess channel formed at each receiver [10] and depicted in Fig. 7(a) as well.

During a fraction of the communication block, we treat the XChannel as a twouser interference channel with swapped IDs in which wishes to communicated with , see Fig. 7(c). In the homogeneous setting of this work, the capacity region of this interference channel with swapped IDs matches that of the previous case and an instance of it is depicted in Fig. 7(c).
Achievable Rates: With this strategy fraction of the times, we achieve a common rate, , of , while fraction of the times, we achieve individual rates
(41) 
The overall achievable rate matches in this case. Now, consider the case in which
(42) 
We need to modify the strategy slightly. The transmission strategy for the interference channel consists of two phases. During Phase 1, uncoded bits intended for different receivers are transmitted. During Phases 2, using the delayed CSIT, the previously transmitted bits are combined and repackaged to create bits of common interest. These bits are then transmitted using the capacityachieving strategy of the multicast problem. In the modified strategy for the XChannel, Phase 1 consists of two subphases. In the first subphase, both transmitters send out bits intended for while in the second subphases, bits intended for are communicated. This way we take full advantage of the entire signal space at each receiver and the individual rates are no longer limited by the capacity of a pointtopoint channel. The second phase is identical to the Interference Channel.
Achieving unequal rates for different users follows the same logic with careful application of subproblems as described in Section VB. Another example is corner point
(43) 
of Fig. 2(a). This corner point corresponds to the MultipleAccess Channel formed at as depicted in Fig. 9.
In this section, we presented the transmission strategy for where . We also discussed and highlighted the strategy for other corner points of the capacity region.
Vi Conclusion
We established the capacity region of finitefield fading XChannels with common messages and delayed CSIT. We presented a new set of outerbounds for this problem that relied on an extremal entropy inequality developed specifically for this problem. We then showed how the outerbounds can be achieved by treating the XChannel as a combination of a number of wellknown problems such the interference channel and the multicast channel.
An important future work is to study Gaussian XChannels with delayed CSIT. One approach could be to extend our results to the multilayer finitefield fading setting similar to [14] and then, derive the capacity region of the Gaussian XChannels to within a constant number of bits.
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